President Trump’s stunning dismissal of John R. Bolton clears the stage for Secretary of State Mike Pompeo — along with Vice President Mike Pence, one of the few members of Mr. Trump’s original national security inner circle still standing — to wield more influence over a raft of foreign policy challenges confronting the White House.

While Mr. Pompeo was publicly aligned with the ousted national security adviser’s notoriously hard-line positions on several fronts, in particular on Iran, the two sharply disagreed in private on a range of other matters, most notably the pursuit of sensitive nuclear negotiations with North Korea.

National security sources said Tuesday that Mr. Bolton’s departure could open the way for renewed “working level” talks and a possible step-by-step approach with North Korea — an approach the State Department was once seen to advocate only to be shut down by Mr. Bolton’s demand that Pyongyang accept an all-or-nothing deal.

It wasn’t the only area where the two men disagreed. Mr. Pompeo’s background as a tea-party-backed star in the House of Representatives has made him more adept at working personally with Mr. Trump, particularly when it comes to threading the policy needle implementing the president’s “America First” foreign policy impulses.

But the secretary of state’s influence was challenged when Mr. Bolton, an entirely different kind of foreign policy conservative, joined the administration’s inner circle in April 2018.

Mr. Bolton’s neoconservative bent — favoring U.S. military intervention and regime change as go-to policy options — initially added heft to the administration’s pressure campaigns against Iran and North Korea. But it also soon clashed with Mr. Trump’s desires to withdraw U.S. forces from undesired conflict zones and to persuade foreign nations to take more responsibility for global security, even if it means pressuring allies to pay more for their own defense.

Mr. Pompeo cultivated a close relationship with Mr. Trump after being named head of the CIA, giving Mr. Trump personal intelligence briefings. One sign of his continuing clout: One of the first names floated as a permanent replacement for Mr. Bolton was Stephen E. Biegun, the secretary of state’s envoy to talks with North Korea and a close political ally.

The secretary of state has not always meshed perfectly with his boss, including Mr. Trump’s surprise recognition of Israeli control of the Golan Heights and the president’s cut of foreign aid programs to Central American nations.

But Mr. Pompeo’s personal rapport with Mr. Trump, his willingness to defend the administration in the media and his determined campaign never to show daylight between himself and the president on major issues are likely to take on new significance with Mr. Bolton gone.

Mr. Pompeo made no secret Tuesday of the fact that he and Mr. Bolton crossed swords on policy over the past year.

“There were many times Ambassador Bolton and I disagreed; that’s to be sure,” the secretary of state told reporters. “But that’s true for lots of people with whom I interact.”

He said separately that Mr. Trump “should have people that he trusts and values and whose efforts and judgments benefit him in delivering American foreign policy.”

The fall of Mr. Bolton and rise of Mr. Pompeo could play out quickly when world leaders gather for the annual U.N. General Assembly in New York this month. While the State Department has energetically implemented Mr. Trump’s “maximum pressure” campaign against Iran after the withdrawal from the 2015 nuclear deal, Mr. Pompeo has been careful to leave room for Mr. Trump’s expressed wish for a possible personal diplomatic overture along the lines of his North Korea policy.

Asked Tuesday whether there could be a precedent-shattering meeting between Mr. Trump and Iranian President Hassan Rouhani at the U.N. gathering, Mr. Pompeo said simply, “Sure.”

“The president has made very clear he is prepared to meet [Iranian leaders] with no preconditions” in search of a better deal to curb Tehran, Mr. Pompeo said.

One Republican insider told The Associated Press that Mr. Bolton’s opposition to such a meeting was a precipitating factor in his dismissal.

Some Republicans appeared to openly cheer Mr. Bolton’s dismissal. Sen. Rand Paul, Kentucky Republican and member of the Senate Foreign Relations Committee, told reporters that Mr. Trump has long been clear that “he’s not for regime change.”

“He said that [on] North Korea, he’s actually said that [on] Iran, and you know, Bolton’s been very, very loud in his call for regime change around the world,” Mr. Paul said. “I think the problem is that it’s a naive point of view to believe that we can militarily topple regimes around the world and that they’ll be replaced with democracies.”

Rep. Jackie Speier, a California Democrat who serves on the House Permanent Select Committee on Intelligence, told CNN that Mr. Trump and Mr. Bolton were like “oil and water,” while the president’s “relationship with Secretary Pompeo is quite strong.”

North Korea diplomacy could feel an immediate shift with Mr. Pompeo’s rising clout. Mr. Bolton infuriated Pyongyang and stunned others in the administration when he said the U.S. was looking for a “Libya-style” model for the North Koreans — even though Libyan dictator Moammar Gadhafi lost his power and then his life after agreeing to give up his nuclear programs in a deal with the George W. Bush administration.

Mr. Bolton’s departure could open the way for more low-key, lower-level contacts between Washington and Pyongyang to better lay the groundwork for the next Trump-Kim summit.

“Bolton’s exit may lead to a more realistic, step-by-step approach to North Korean denuclearization, where Bolton’s ‘all or nothing’ approach to easing sanctions has produced a deadlock despite three Trump-Kim summits,” said Alexander Vershbow, a former U.S. ambassador to Russia and South Korea and a former NATO deputy secretary-general.

“Bolton’s departure was only a matter of time,” Mr. Vershbow, a fellow with the Atlantic Council’s Scowcroft Center for Strategy and Security, said in comments circulated to reporters. “Still, the timing is ironic, coming after Trump’s termination of negotiations with the Taliban on which Bolton was right to be skeptical (maybe Trump didn’t like hearing ‘I told you so’).”

Mr. Pompeo has also apparently been spared the occasional dressing-down that former Trump aides say are a regular feature of working in this administration.

“I argue with everyone,” Mr. Trump told New York Magazine in a 2018 interview. “Except Pompeo. I don’t think I’ve had an argument with Pompeo.”